No, you don’t need to include your address, or the address of the hiring manager anymore. In the olden days when cover letters were sent in the mail or submitted in person, they included both the applicant’s and the recipient’s addresses. However, cover letters are now exclusively submitted electronically, so an email address and phone number is all that is needed.
Additionally, if you’re applying for a job in a different region, state or country, including an address on your cover letter (or resume) could put you at a disadvantage when you are being compared to local candidates.